Olaf Lechtenfeld and Alexander D. Popov-Supertwistors and Cubic String Field Theory for Open N=2 Strings

8/3/2019 Olaf Lechtenfeld and Alexander D. Popov-Supertwistors and Cubic String Field Theory for Open N=2 Strings

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arXiv:h

ep-th/0406179v2

9Aug2004

hep-th/0406179ITPUH19/04

Supertwistors and Cubic String Field Theory

for Open N=2 Strings

Olaf Lechtenfeld and Alexander D. Popov

Institut fur Theoretische Physik, Universitat Hannover

Appelstrae 2, D-30167 Hannover, Germany

lechtenf, popov @itp.uni-hannover.de

Abstract

The known Lorentz invariant string field theory for open N=2 strings is combined with a gener-alization of the twistor description of anti-self-dual (super) Yang-Mills theories. We introduce aChern-Simons-type Lagrangian containing twistor variables and derive the Berkovits-Siegel co-variant string field equations of motion via the twistor correspondence. Both the purely bosonicand the maximally space-time supersymmetric cases are considered.

On leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia
http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2http://arxiv.org/abs/hep-th/0406179v2


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1 Introduction

It was recently shown by Witten [1] that B-type open topological string theory with the supertwistorspace CP3|4 as a target space is equivalent to holomorphic Chern-Simons (hCS) theory on the samespace (for related works see [2][7]). This hCS theory in turn is equivalent to supersymmetric N=4anti-self-dual Yang-Mills (ASDYM) theory in four dimensions. The N=4 super ASDYM model isgoverned by the Siegel Lagrangian [8]. Its truncation to the bosonic sector describes N=0 ASDYMtheory with an auxiliary field of helicity 1 [8, 9].

It may be of interest to generalize the twistor correspondence to the level of string field theory(SFT). This could be done using the approach proposed in [5] or in the more general setting of [6].Alternatively, one could concentrate on (an appropriate extension of) SFT for N=2 string theory.At tree level, open N=2 strings are known to reduce to the ASDYM model in a Lorentz noninvariantgauge [10]; their SFT formulation [11] is based on the N=4 topological string description [12, 13].The latter contains twistors from the outset: The coordinate CP1, the linear system, theintegrability and the classical solutions with the help of twistor methods were all incorporatedinto N=2 open string field theory in [14, 15]. Since this theory [11] generalizes the Wess-Zumino-

Witten-type model [16] for ASDYM theory and thus describes only anti-self-dual gauge fields(having helicity +1), it is not Lorentz invariant. Its maximally supersymmetric extension, N=4super ASDYM theory, however, does admit a Lorentz-invariant formulation [8, 9]. This theory andits truncation to N=0 features pairs of fields of opposite helicity. In [17] it was proposed to lift thecorresponding Lagrangians to SFT.

In the present paper the twistor description of both the purely bosonic and the N=4 supersym-metric ASDYM models [8, 9] is raised to the SFT level. In contrast to previous proposals [1][7],we allow the string to vibrate only in part of the supertwistor space. The remaining coordinates ofthis space are not promoted to word-sheet fields but kept as non-dynamical string field parameters.Concretely, we propose a cubic action containing an integration over the supertwistor space CP3|4

and show that its hCS-type equations of motion are equivalent to the covariant string field equa-

tions introduced in [17]. This model may be regarded as a specialization of Wittens supertwistorSFT and may even be equivalent to it. In any case, it is directly related with N=4 super ASDYMtheory in four dimensions. We also consider its proper truncation to the b osonic sector, whichyields a twistor SFT related to non-supersymmetric ASDYM theory.

2 Covariant string field theory for open N=2 strings

Open N=2 strings. From the worldsheet point of view critical open N=2 strings in a flat four-dimensional space-time of signature ( + +) or (+ + + +) are nothing but N=2 supergravity on atwo-dimensional (pseudo) Riemannian surface with boundaries, coupled to two chiral N=2 matter

multiplets (X, ). The latters components are complex scalars (the four imbedding coordinates)and Dirac spinors (their four NSR partners) in two dimensions. In the ghost-free formulation ofthe N=2 string one employs the extension of the c=6, N=2 superconformal algebra to the smallN=4 superconformal algebra1

T = zXzX +

z ,

G = zX , J

= ,

(1)

1We raise and lower indices with 12 = 21 = 1, 12 = 21 = 1, and similarly for , and

, .

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where , = 1, 2 and , = 1, 2 are space-time spinor indices and , = 1, 2 denote the world-sheet internal indices associated with the group SU(1,1) (Kleinian space R2,2) or SU(2) (Euclideanspace R4,0) of R symmetries. For the reality structures imposed on target space coordinates andsuperconformal algebra generators see [12, 11, 18, 15]. After twisting this algebra,

D := G1

(2)

become two fermionic spin-one operators which subsequently serve as BRST-like currents sincethey are nilpotent [12, 11],

(D1)2 = 0 = (D2)

2 and

D1 , D2

= 0 . (3)

Furthermore, 1 is now conformal spin zero while 2 is conformal spin one.

Covariant string field theory. Following Berkovits and Siegel [17], we introduce two Lie-algebravalued fermionic string fields A[X, ] and three Lie-algebra valued bosonic string fields G

[X, ](symmetric in and ). Although we suppress it in our notation, string fields are always multipliedusing Wittens star product (midpoint gluing prescription) [19].2 The index structure reveals thatthe fields G parametrize the self-dual

3 tensor G, = G on the target space.

The Lorentz invariant string field theory action [17] reads

SBS = tr (GF) , (4)

where . . . means integration over all modes of X and , the trace tr is taken over the Liealgebra indices and

F := DA + DA +

A , A

. (5)

Note that the action of D on any string field B is defined in conformal field theory language astaking the contour integral [11]

DB

(z) =

zdw

2 iD(w) B(z) . (6)

The covariant string field equations of motion following from the action (4) read

F = 0 and DG +

A , G

= 0 . (7)

For a supersymmetric generalization of the action (4) Berkovits and Siegel [17] introduce amultiplet of string fields

A , i , ij , i , G

with i, j = 1, 2, 3, 4 (8)

imitating the N=4 ASDYM multiplet [8]. Here, A and i are fermionic while i, ij and G

are bosonic. Ref. [17] proposes the following action for this super SFT:

SBS = tr GF + 2

i i +18

ijklij kl +

12

ijklij k l (9)withB := DB + AB (1)

|B|BA , (10)

where the Grassmann parity |B| equals 0 or 1 for bosonic or fermionic fields B, respectively. Dueto the large number of string fields this model seems unattractive. However, as we shall see in thecoming section, all these fields appear as components of one string field living in a twistor extendedtarget space.

2The star product was concretized in oscillator language for bosons in [20] and for twisted fermions in [21].3Self-duality can always be interchanged with anti-self-duality by flipping the orientation of the four-dimensional

target space. For the choice of the orientation made in [1, 22] these G parametrize a self-dual tensor.

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3 Cubic string field theory for open N=2 strings

Supertwistor space notation. In the Appendix we describe the supertwistor space P3|4 of

the space (R4, g) with the metric g = diag(, , 1, 1) and = 1. It is fibered over the realtwo-dimensional space with 1 = CP

1 and +1 = H2 covered by two patches U. The space

P

3|4

is parametrized by four even complex coordinates (x

) C4

subject to the reality conditionsx22 = x11 and x21 = x12, complex coordinates U and odd (Grassmann) coordinates

i,

i = 1, . . . , 4. The space P3|4 is a Calabi-Yau supermanifold [1]. From now on we shall work on the

patch U+ of , and for notational simplicity we shall omit the subscript + in + U+,

i+ etc.

For further use we introduce

() =1

, () =

1

, () =

and = (1 )1 . (11)

BRST operator. Let us introduce the operator

D := D = 1 z( X) (12)

taking values in the holomorphic line bundle O(1). We notice that the operators zX act as

derivatives on string fields. Their zero mode parts, x

, form two type (0,1) vector fields on

the bosonic twistor space P3 fibered over (see the Appendix for more details). Recall that P3

being an open subset ofCP3 is the twistor space of (R4, g). In order to obtain a general type (0,1)vector field along the twistor space one should therefore extend the operator (12) by adding thetype (0,1) derivative along ,

:= d

. (13)

Assuming that string fields now depend on the extra variable , we define the operator

Q := D + (14)

acting on string fields via (6) for the D part and by ordinary differentiation with respect to .It is easy to see that Q2 = 0 due to (3) and the facts that D does not depend on and that

{d, } = 0 (cf. [23]). We take this nilpotent operator as the BRST operator of our SFT

extended to 1|4 P

3|4 .

String fields. We now consider a fermionic (odd) string field A[X,,i, , ] depending not onlyon X() and () but also on i and on the parameter . It is important to realize that i

and do not depend on here but may be considered as zero modes of world-sheet fields. Since the

operator Q has the split form (14) it is natural to assume the same splitting of the string field A,

A = AD[X,,i, , ] + A[X, ,

i, , ] with A = Ad , (15)

where AD gauges D and A gauges . Note also that D takes values in O(1) and in O(0);therefore, AD and A are O(1) and O(0) valued, respectively. Since d is the basis section of the

bundle O(2) complex conjugate to O(2) and anticommutes with spinors we reason that Ais bosonic (even) and takes values in O(2). It is also assumed that a term Ad is absent in thesplitting (15), i.e. A is a string field of the (0,1)-form type (cf. [1] for the B-model argument). Notethat by definition the string field AD does not contain d and is fermionic (odd).

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Cubic action. Having d and di we introduce the action

S =

d

d1 d2 d3 d4 tr

A QA + 23 A

3

, (16)

where . . . is the same integration over (X, ) modes as in (4). Note that

d acts as integration

over

for terms containing d and as a contour integral around =0 for other terms. TheLagrangian L in (16) can be split into two parts,

L = tr

A QA + 23 A3

= L1 + L2 , with (17)

L1 = tr

AD AD + 2 AD DA + 2 AA2D

, (18)

L2 = tr

AD DAD +23 A

3D

, (19)

where we used the cyclicity under the trace and omitted total derivatives.

It is important to note that L1 takes values in O(2), which is compensated by the holomorphicmeasure dd1 d2 d3 d4 being O(2) valued.4 At the same time, L2 takes values in O(3) which

causes it to drop out of the action by virtue of Cauchys theorem applied to the contour integral.Thus, the action (16) can be rewritten as

S =

d

d1 d2 d3 d4 tr

AD AD + 2 AD DA + 2 AA

2D

. (20)

Moreover, both forms (16) and (20) of the action lead to the same Chern-Simons-type equation ofmotion,

Q A + A2 = 0 (21)

which decomposes into

DA + AD + AD , A = 0 (22)and DAD + A2D = 0 . (23)Component analysis. Recall that AD and A take values in the bundles O(1) and O(2),

respectively. Together with the fact that the i are nilpotent and O(1) valued, this determines thedependence of AD and A = Ad on

i, and . Namely, this dependence has the form (cf. [22])

AD = A +

ii +2!

ij ij +2

3! ijk ijk +

3

4! ijklGijkl ,

A =2

2! ijij +

3

3! ijk ijk +

4

4! ijklGijkl ,

(24)

where , and are given in (11) and i1i2...ik := i1i2 . . . ik . The expansion (24) is defined upto a gauge transformation generated by a group-valued function which may depend on and . Allstring fields appearing in the expansion (24) depend only on X() and (). From the propertiesof AD, A and

i it follows that the fields with an odd number of spinor indices are fermionic(odd) while those with an even number of spinor indices are bosonic (even). Moreover, due to thesymmetry of the products and the skewsymmetry of the i products all component fields areautomatically symmetric in their spinor indices and antisymmetric in their Latin indices.

4The choice of four Grassmann coordinates i is dictated by the Calabi-Yau condition: The contribution of thecoordinates (X,, ) to the first Chern number is (2, 2,4), respectively.

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Substituting (24) into (22), we obtain the equations5

ij = ij and ijk =12 (

)ijk and Gijkl =

13 (G)ijkl (25)

showing that (ij , ijk, Gijkl) is a set of auxiliary fields. The other nontrivial equationsfollowing from (22) and (23) after substituting (24) read6

F DA + DA + A , A = 0 ,i = 0 ,

i + 2

ij , j

= 0 ,

ij + 2

i , j

= 0 ,

G + 2

i , i

+

ij ,

ij

= 0 ,

(26)

where we introduced

ij := 12! ijklkl and

i :=

13!

ijkljkl and G :=

14!

ijklGijkl . (27)

Up to constant field rescalings

G G , i 12

i , ij 12 ij and

i

i (28)

the equations (26) for =1 coincide with the equations of motion following from the action (9)proposed by Berkovits and Siegel [17]. In the zero mode sector they reduce to the anti-self-dual N=4super Yang-Mills equations of motion. Hence, we have established that maximally supersymmetricASDYM theory can be obtained from the standard cubic SFT for a single string field A afterextending the setting to the supertwistor space.

4 Bosonic truncation of open string field theory

In order to make contact with non-supersymmetric ASDYM theory, we subject our string field Afrom (15) and (24) to the truncation conditions

d1 d2 d3 d4

iij

ijk

A = 0 . (29)

These conditions imply that A depends only on the combination

:= 1234 , i.e. A = A[X, ,,, ] . (30)

Obviously, the even nilpotent variable belongs to the bundle O(4) and the integration measurein (29) to O(4).

The properties of the truncated string field

A = AD[X, ,,, ] + A[X,,,, ] (31)5Round brackets denote symmetrization with respect to enclosed indices.6Recall that = 1 for signature (++) and = 1 for signature (++++).

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are the same as before the truncation, except for the restricted dependence on the Grassmannvariables. The operators Q, D and , the actions (16) and (20), the Lagrangians (17)(19) and theequations of motion (21)(23) are unchanged. However, the expansion (24) now simplifies to

AD = A[X, ] +

3G[X, ] ,

A = 4

G[X, ] d ,

(32)

where (see (27))

G =14!

ijklGijkl and G :=14!

ijklGijkl . (33)

From the properties of AD and A it follows that the string fields A and G are odd and theG are even.

Substituting the expansion (32) into the equations of motion (22) and (23), we recover for Aand G the bosonic string field equations

F = 0 and DG + A , G

= 0 (34)displayed already in (7) and for G the dependence

G = 13 (G) (35)

as expected. The same result occurs when putting to zero in (26) the string fields i, i and ij

as the truncation (29) demands. All other equations following from (22) and (23) are satisfiedautomatically, due to (34) and the Bianchi identities.

Hence, we have proven that the cubic supertwistor SFT defined by the action (16) togetherwith the geometric truncation conditions (29) is equivalent to the Berkovits-Siegel SFT given bythe action (4). Moreover, (4) and (9) derive from (16) simply by substituting there the expansion

(32) or (24), respectively, and integrating over the Grassmann and twistor variables. All this issimilar to the field theory case [1, 22] where in the supertwistor reformulation of N=4 ASDYMtheory as hCS theory the dependence of all fields on the twistor variable is fixed (up to a gaugetransformation) by the topology of the supertwistor space and one can integrate over it, descendingfrom six to four real dimensions.

5 Conclusions

The basic result of this paper can be summarized in the equations

S = dd1 d2 d3 d4 tr A QA + 23 A3 (36)=

dd

d1 d2 d3 d4 tr

AD AD + 2 AD DA + 2 A

2D

A

(37)

= c tr

GF + 2 i i +

14

ij ij +

ij k

l

, (38)

where c is an inessential numerical constant. The first step demands a split A = AD + Ad of

the basic (supertwistor) string field. The second step requires integrating over 1|4 and rescaling

the field as in (28). Truncating A to its lowest and highest Grassmann components (the O(0) and

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O(4) parts) projects the above action to tr(GF), which governs bosonic N=2 open SFT.Finally, reducing to the string zero modes one recovers the twistor description of N=4 and N=0ASDYM on the field theory level.

Acknowledgements

This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG).

A Appendix: Supertwistor space

The twistor space ofR4,0. Let us consider the Riemann sphere CP1 = S2 with homogeneouscoordinates () C2. It can be covered by two patches,

U+ =

(1, 2) : 1 = 0

and U =

(1, 2) : 2 = 0

(39)

with coordinates + := 2/1 on U+ and := 1/2 on U. On the intersection U+ U we have+ =

1 .

The holomorphic line bundle O(n) over CP1 is defined as a two-dimensional complex manifoldwith the holomorphic projection

: O(n) CP1 (40)

such that it is covered by two patches U+ and U with coordinates (w+, +) on U+ and (w, )on U related by w+ =

n+w and + =

1 on U+ U. A global holomorphic section of O(n)

exists only for n0. Over U CP1 it is represented by polynomials p(n) in of degree n with

p(n)+ =

n+p

(n) on U+ U.

Recall that the Riemann sphere

CP1 = SO(4)/U(2) (41)

parametrizes the space of all translational invariant (constant) complex structures on the Euclideanspace R4,0, and the space P3E := R

4CP1 is called the twistor space ofR4,0 [24]. As a complexmanifold P3E is a rank 2 holomorphic vector bundle O(1) O(1) over CP

1:

P3E = O(1) O(1) . (42)

For more details and references see e.g. [24, 25, 22].

The twistor space ofR2,2. In the Kleinian space R2,2 of signature ( + +) constant complexstructures are parametrized by the two-sheeted hyperboloid

H2 = H+ H = SO(2, 2)/U(1, 1) , (43)

whereH+ =

+ U+ : |+| < 1

= SU(1, 1)/U(1)

and H =

U : || < 1

= SU(1, 1)/U(1) .(44)

In fact, under the action of the group SU(1,1) the Riemann sphere is decomposed into three orbits,CP1 = H+ S1 H, where the boundary of both H+ and H is given by

S1 =

CP1 : || = 1

= SU(1, 1)/B+ with B+ =a1+ i a2 a3 i a2

a3+ i a2 a1 i a2

(45)

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with a1,2,3 R and a21a22 = 1.

The twistor space ofR2,2 is the space P3K := R4H2 which as a complex manifold coincides

with the restriction of the rank 2 holomorphic vector bundle (42) to the bundle over H2 CP1.Equivalently, it can be described as a space P3K = P

3E \ T

3, where T3 is a real three-dimensionalsubspace of P3E stable under an anti-linear involution (real structure) which can be defined on P

3E.

For more details see e.g. [26].

Vector fields of type (0,1). For considering both signatures together, we denote by thespace of complex structures on (R4, g) with the metric g = diag(, , 1, 1) and = 1, so that

1 = CP1 and +1 = H

2 . (46)

Therefore, is covered by two patches U with U

1 = U and U

+1 = H. Analogously, we

denote by P3 the twistor space of (R4, g) with P31 = P

3E and P

3+1 = P

3K. The complex manifold

P3 is covered by two patches V with complex coordinates (w

, ) on V

. We introduce

( ) = 1

, () =

1

, + = (1 ++)

1 and = (1 )1 (47)

for U. Note that in terms of ( ) or () a section of the bundle O(n) over U can bewritten as

p(n) = p

1...n 1 . . . n = p1...n

1 . . .

n . (48)

Recall that (R4, g) can be parametrized by coordinates (x) C4 with the reality conditions

x22 = x11 and x21 = x12 [22]. On the twistor space P3= R4 we have coordinates (w, w

3) =

(x , ) or (x, , ). The antiholomorphic vector fields /w

and /w

3 can be rewritten

in the coordinates (x, , ) as

w1=

x2,

w2=

x1and

w3=

x1

x. (49)

The vector fieldsv =

xand v

3=

(50)

can be taken as a basis of vector fields of type (0,1) on P3 .

Supertwistors. Let us now add four odd variables i such that

i j + j i = 0 for i, j = 1, 2, 3, 4 (51)

and each i takes its value in the line bundle O(1) over CP1 (cf. [1]). For describing them formallywe introduce a Grassmann parity changing operator which, when acting on a vector bundle, flipsthe Grassmann parity of the fibre coordinates. Hence, we consider the bundle C4 O(1) CP1

which is parametrized by complex variables U CP1 and fibre Grassmann coordinates isuch that i+ = +i on the intersection of the two patches covering the total space of this vectorbundle.

With the Grassmann variables i one can introduce the supertwistor space P3|4E as a holomorphic

vector bundle over CP1, namely

P3|4E = C

2 O(1) C4 O(1) . (52)

The supertwistor space P3|4K is defined as a restriction of the bundle P

3|4E CP

1 to the bundleover the two-sheeted hyperboloid H2 CP1.

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References

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Olaf Lechtenfeld and Alexander D. Popov-Supertwistors and Cubic String Field Theory for Open N=2 Strings